Optimal quantum cloning based on the maximin principle by usinga prioriinformation

Abstract

We propose an optimal $1\ensuremath{\rightarrow}2$ quantum cloning method based on the maximin principle by making full use of a priori information of amplitude and phase about the general cloned qubit input set, which is a simply connected region enclosed by a ``longitude-latitude grid'' on the Bloch sphere. Theoretically, the fidelity of the optimal quantum cloning machine derived from this method is the largest in terms of the maximin principle compared with that of any other machine. The problem solving is an optimization process that involves six unknown complex variables, six vectors in an uncertain-dimensional complex vector space, and four equality constraints. Moreover, by restricting the structure of the quantum cloning machine, the optimization problem is simplified as a three-real-parameter suboptimization problem with only one equality constraint. We obtain the explicit formula for a suboptimal quantum cloning machine. Additionally, the fidelity of our suboptimal quantum cloning machine is higher than or at least equal to that of universal quantum cloning machines and phase-covariant quantum cloning machines. It is also underlined that the suboptimal cloning machine outperforms the ``belt quantum cloning machine'' for some cases.